Sounds like you are trying to find an "intrinsic" price-to-book ratio of some sort? At first glance, we're not sure why this is relevant. P/B is a market multiple; so there is no "intrinsic" or "correct" answer, but rather, what investors (the market) are willing to pay. This changes over time and is subject to the markets.
Then, upon further consideration of your approach, we thought about it again to try to see your logic:
ROE is calculating the margin, with equity (book value) in the denominator
Cost of equity (via CAPM) is calculating the required return investors demand
So, in trying to see some logic, if we can somehow extrapolate cost of equity to somehow get a price in the numerator to cross multiply something to get price/book ratio.....
We pose this question back to you:
What if ROE is only 6% and CAPM says 12% return, both of which are not unreasonable.
According to your logic, that would imply a P/B ratio of 0.5x.
So we draw the conclusion that market price (numerator of P/B ratio) cannot be concluded from dividing ROE by cost of equity.